Triangular Nim, one variant of the game Nim, is a common two-player game in Taiwan
and China. In the past, Hsu [9] strongly solved seven layer Triangular Nim while some of
the authors recently strongly solved eight layer Triangular Nim. The latter required 8
gigabytes in memory and 8,472 seconds. Using a retrograde method, this paper strongly
solves nine layer Triangular Nim. In our first version, the program requires four terabytes
in memory and takes about 129.21 days aggregately. In our second version, improved by
removing some rotated and mirrored positions, the program reduces the memory by a factor
of 5.72 and the computation time by a factor of 4.62. Our experiment result also shows that
the loss rate is only 5.0%. This is also used to help improve the performance.

Received February 26, 2011; revised August 19, 2011; accepted August 29, 2011.
Communicated by Yuh-Jye Lee.
* This work was supported by the National Science Council of Taiwan, under contracts No. NSC 97-2221-E-009-126-MY3 and NSC 99-2221-E-009-102-MY3.